PhysicsLAB AP Free Response Question
2009 B1

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In an experiment, students are to calculate the spring constant k of a vertical spring in a small jumping toy that initially rests on a table. When the spring in the toy is compressed a distance x from its uncompressed length Lo and the toy is released, the top of the toy rises to a maximum height h above the point of maximum compression.
The students repeat the experiment several times, measuring h with objects of various masses taped to the top of the toy so that the combined mass of the toy and added objects is m. The bottom of the toy and the spring each have negligible mass compared to the top of the toy and the objects taped to it.
 (a) Derive an expression for the height h in terms of m, x, k, and fundamental constants.

With the spring compressed a distance x = 0.020 meters in each trial, the students obtained the following data for different values of m.
 i. What quantities should be graphed so that the slope of a best-fit straight line through the data points can be used to calculate the spring constant k?

 ii. Fill in one or both of the blank columns in the table with calculated values of your quantities, including units.

(c) On the axes below, plot your data and draw a best-fit straight line. Label the axes and indicate the scale.

 (d) Using your best-fit line, calculate the numerical value of the spring constant.

 (e) Describe a procedure for measuring the height h in the experiment, given that the toy is only momentarily at that maximum height.

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